Scalar field critical collapse in axisymmetry

with Matt Choptuik, Eric Hirschmann and Steve Liebling

The results presented here were obtained with a 2D axisymmetric adaptive code, featuring:

Berger and Oliger (B&O) adaptive mesh refinement (AMR)
Truncation error estimation using a self-shadow hierarchy
Equations derived using the 2+1+1 formalism, in cylindrical (rho,z) coordinates
Axis instabilities eliminated by applying appropriate regularity conditions to all variables, and adding Kreiss-Oliger dissipation to evolution equations
Constrained evolution --- the elliptics are incorporated into the B&O scheme by using a particular form of extrapolation in conjunction with post-injection-solution of the elliptic equations (via multigrid)
Current matter sources are minimally-coupled, real and complex scalar fields

Movies

Massless real scalar field

All animations below are of the scalar field, where the height and color of the image encode the magnitude of the field. Log coordinates show the solution transformed to (ln(r+epsilon),theta) coordinates, while rho-z coordinates are the untransformed coordinates of the simulation.

Spherically symmetric initial data

weak field example, rho-z coords: phi_lo (3.1MB)
black hole forming example, rho-z coords: phi_hi (2.5MB)
near-critical solution, rho-z coordinates: phi_crit_unif (1.6MB)
near-critical solution, log coords, logarithmic spacing in time (and a rotated view) : phi_crit_log (4.6MB)

Prolate initial data

The following movies demonstrate an apparent "focusing" instability of the spherical critical solution to axisymmetric perturbations. The initial amplitude of this putative second unstable mode seems to be closely related to the prolateness of the initial data, as demonstrated by the following sequence of animations (e² is the prolateness factor as defined in Critical Collapse of the Massless Scalar Field in Axisymmetry: e²=0 denotes a spherical distribution, e²=1 a cylindrical one)

e² = 0 (i.e., spherically symmetric, but different view from the one above), log coords: phi_pr_1_0 (3.5MB)
e² = 1/2, log coords: phi_pr_1_2 (4.5MB)
e² = 3/4, log coords: phi_pr_1_4 (3.9MB)
e² = 5/6, log coords: phi_pr_1_6 (5.1MB), smaller version phi_1_6 (3.2MB)
e² = 7/8, log coords: phi_pr_1_8 (5.1MB)

"Anti-symmetric" initial data

The initial data for this simulation is an imploding pulse of scalar field energy, anti-symmetric about z=0. The anti-symmetry prevents any echoing solution from developing at z=0 (the center of the implosion) at threshold. Instead, two local self-similar solutions develop off center. Here is an animation of the scalar field phi in (rho,z) coordinates: phi.mpg (3.9 MB) (smaller version: phi_small.mpg (2.43MB)). The color and height of the surface in the animation represents the magnitude of the scalar field.

The above solution was tuned to threshold to within 1 part in 10^16. The anti-symmetry is not preserved exactly during evolution due to numerical errors, and hence closer than around 10^-8 to threshold only the left-hand (+z) echoing solution could be tuned to criticality. Around 3 full echoes are observed in the +z solution, though the self-similar nature of the solution prevents one from seeing this in the cylindrical coordinates used in the animation. The following image, phi_ln(r).jpeg, is a snapshot of the last time step of the simulation transformed to a radial logarithmic coordinate r and angular coordinate theta, to better demonstrate the self-similar nature of the solution. The transformation is centered about the +z solution, and so the -z solution is severely distorted in the image.

To illustrate the kind of grid hierarchies produced during evolution, the following series of images shows phi (in rho,z coordinates) as a 2:1 coarsened wire frame mesh at the initial time: phi_wf_0.jpeg, and at the latest time, with each successive frame zooming into the region of more refinement: phi_tf_z1.jpeg, phi_tf_z2.jpeg, phi_tf_z3.jpeg, phi_tf_z4.jpeg, phi_tf_z5.jpeg, phi_tf_z6.jpeg. (The base grid has a resolution of 65x129, and up to 24 levels of 2:1 refinement were used during evolution.)

Complex Scalar Field

Spherical initial wavefront

Near critical solution, from an initial spherical wavefront (though the energy distribution is not, and never can be spherically symmetric, because of the "m=1" azimuthal dependence of the field) with zero net angular momentum :

real component of scalar field, viewed from above in (rho,z) coordinates (i.e., the z-direction is up, and the rho-direction is to the right): phi3_r_rho (2.2MB)

the same field, but transformed to (ln(r+epsilon),theta) coordinates, and shown from a different viewpoint: lnr_phi3_r_rho (6.6MB)

Net angular momentum appears to play an insignificant role in the threshold solution of similar initial data with angular momentum, i.e. it looks the same, and so I'm not including such animations for now.

Critical solution parameters:

Scaling exponent gamma ~ 0.11
Echoing exponent delta ~ 0.42

A preliminary estimate of the scaling exponent for the net angular momentum J of black holes of mass M that form in supercritical collapse suggests J ~ M⁶(possibly even a higher power of M). This estimate comes from measuring initial apparent horizon properties rather far from threshold, and hence is uncertain. However, the net J becomes small so rapidly as one approaches the critical solution in parameter space that it will be almost impossible to obtain an accurate estimate of it close to threshold.

Prolate initial wavefront

An early result from near-critical collapse of a prolate initial wavefront (with a prolateness factor e²=1/2 ... see the massless scalar field results above). The critical solution appears to be subject to a similar "focusing instability" that may be present for the massless, real threshold solution. This initial distribution does have net angular momentum (J/M^2 ~ 0.2, where M is the ADM mass), however, as with the spherical wavefronts angular momentum appears to play an insignificant role at threshold.

in (rho,z) coordinates viewed from above: pr_12_phi3_r_rho (4.0MB)

transformed to a logarithmic radial coordinate: lnr_pr_12_phi3_r_rho (6.3MB)

Note that a different scale and colormap range was used compared to the spherical wavefront animations above. The sequence of output times was chosen assuming a single "accumulation point" for the critical solution, which is strictly not valid for this solution. Also, the output time changes to uniform after the critical behaviour to show the dispersal, hence the speed up in the last few frames.

last updated: Oct. 2003